Lexicographic ordering of Pauli strings has been shown to be effective at enabling gate cancellation between them [1]}, [2]}. Here we adapt this principle to the multi-string-per-block case for our gate-count-oriented scheduling algorithm.
In the lexicographic order, the Pauli strings are scheduled in the alphabetical order.
In Figure REF , we assume \(X < Y < Z < I\)  and use little-endian to lexicographically order from \(q7\)  down to \(q0\) .
When a block has multiple strings, we first apply the lexicographic order on all strings in this block and then use the first string to represent this block when compared with other blocks.
The first Pauli string can be representative because the strings in one block are usually mutually commutative in practical algorithmic constraints [3]}, [4]}, [2]}.
Two strings in a mutually commutative set can share the same operators on many qubits and all strings in one block are similar.
Figure REF  (b) shows the result of gate-count-oriented scheduling.
